A Few Facts About Quasiconformal Mappings
If then the maps and are both quasiconformal and have constant dilatation .
If then the map is quasiconformal (here is a complex number) and has constant dilatation . When, this is an example of a quasiconformal homeomorphism that is not smooth. If, this is simply the identity map.
A homeomophism is 1-quasiconformal if and only if it is conformal. Hence the identity map is always 1-quasiconformal. If is K-quasiconformal and is K' -quasiconformal, then is K K' -quasiconformal. The inverse of a K-quasiconformal homeomorphism is K-quasiconformal. Hence the set of quasiconformal maps forms a group under composition.
The space of K-quasiconformal mappings from the complex plane to itself mapping three distinct points to three given points is compact.
Read more about this topic: Quasiconformal Mapping
Famous quotes containing the word facts:
“Science is facts. Just as houses are made of stones, so is science made of facts. But a pile of stones is not a house and a collection of facts is not necessarily science.”
—Jules Henri Poincare (1854–1912)